The present invention relates to digital filters, and more particularly to digital finite impulse response (FIR) filters for compensation of the Nyquist slope in pseudo-synchronous demodulators.
If two arbitrary baseband sinusoids of frequencies .omega..sub.h (high frequency) and .omega..sub.l (low frequency) EQU V[t]=A Cos [.omega..sub.h t]+B Cos [.omega..sub.l t] (1)
are modulated with another sinusoid at angular frequency .omega..sub.m, Sin[.omega..sub.m t], the modulated signal, as shown in FIG. 1, is: EQU M[t]=A Sin [.omega..sub.m t]+B Sin [.omega..sub.m t] Cos [.omega..sub.l t]=(A/2) (Sin [(.omega..sub.m -.omega..sub.h)t]+Sin [(.omega..sub.m +.omega..sub.h)t])+(B/2) (Sin [(.omega..sub.m -.omega..sub.l)t]+Sin [(.omega..sub.m .omega..sub.l)t]). (2)
Using a highpass filter to remove the terms while passing the (.omega..sub.m -.omega..sub.h)terms while passing the (.omega..sub.m +.omega..sub.h) terms also provides 3 db down response at .omega..sub.m as shown in FIG. 2 and, since .omega..sub.l is close to zero, B is halved and Sin[.omega..sub.m -.omega..sub.l t] is approximately equal to Sin[[(.omega..sub.m +.omega..sub.l ]. Multiplying by two produces the final modulated signal: EQU M[t]=A Sin [(.omega..sub.m +.omega..sub.h)t]+B Cos [.omega..sub.l t] Sin [.omega..sub.m t]. (3)
In other words the low frequency input at frequency .omega..sub.l passes through without a quadrature component.
In a synchronous demodulation system the modulated signal is remodulated with Sin[.omega..sub.m t]: EQU M[t] Sin [.omega..sub.m t]=A Sin [(.omega..sub.m +.omega..sub.h)t] Sin [.omega..sub.m t]+B Cos [.omega..sub.l t] Sin [.omega..sub.m t] 2= EQU (A/2) (Cos [(.omega..sub.m .omega..sub.h)t-.omega..sub.m t]-Cos [(.omega..sub.m+.omega..sub.h)t+.omega..sub.m t]+B Cos [.omega.]t](1/2)(1-Cos [2.omega..sub.m t])= EQU (A/2) (Cos [.omega..sub.h t]-Cos [(2.omega..sub.m .omega..sub.h)t]+(B/2) (Cos [.omega..sub.l t](1-Cos [2.omega..sub.m t]). (4)
After filtering and removing components near and above 2.omega..sub.m, and scaling by two, the synchronous in-phase component is: EQU Is[t]=A Cos [.omega..sub.h t]+B Cos [.omega..sub.l t]=V[t] (5)
which is the original signal.
Now demodulating synchronously in quadrature: EQU M[t] Cos [.omega..sub.m t]=A Sin [(.omega..sub.m +.omega..sub.H)t] Cos [.omega..sub.m t]+B Cos [.omega..sub.l t] Sin [.omega..sub.m t] Cos [.omega..sub.m t] EQU =(A/2) (Sin [(.omega..sub.m .omega..sub.h)t-.omega..sub.m t]+Sin [(.omega..sub.m .omega..sub.h)t+.omega..sub.m t])=+(B/2) Cos [.omega..sub.l t](1-Cos [2.omega..sub.m t])= EQU =(A/2) (Sin [(.omega..sub.h t]+Sin [(2.omega..sub.m +.omega..sub.h)t])+(B/2) (Cos [.omega..sub.l t] Sin [2.omega..sub.m t].(6)
After filtering and removing components near and above 2.omega..sub.m, and scaling by two, the B term disappears and the quadrature component is: EQU Qs[t]A Sin [.omega..sub.h t]=A Cos [.omega..sub.h t-.pi./2](7)
resulting in no quadrature component for the low frequencies and a ninety degree phase lag at all higher frequencies with respect to Is[t]. This is undesirable, so two approaches have been used to address this problem.
Synchronous decoding requires using a phase-locked loop (PLL) to lock onto the demodulated signal above, which is extremely difficult due to the narrow bandwidth required. For video signals, as shown in FIG. 3, there are sync areas in which the higher frequency video components .omega..sub.h are absent, allowing a close approximation to the picture carrier .omega..sub.m to be obtained. This has the advantage of allowing the PLL bandwidth to be adjustable so that signal jitter may be observed. However obtaining very fast PLL response is difficult, given that the video sync tip occurs once per video line, approximately 15 kHz, and the PLL bandwidth has to be less than this, typically about 5 kHz. Thus the system does not give best demodulation results EQU I(t)=Cos (.phi.)Is(t)+Sin (.phi.)Qs(t) (8) EQU Q(t)=-Sin (.phi.)Is(t)+Cos (.phi.)Qs(t) (9)
for signals with high-frequency jitter.
A pseudo-synchronous, or quasi-synchronous, demodulation technique using the side-band symmetry near the picture carrier .omega..sub.m applies a bandpass filter to the modulated signal around .omega..sub.m, as shown in FIG. 4, such that A=0. Therefore for small .omega..sub.l : EQU Is'(t)=B Cos [.omega..sub.l t] (10) EQU Qs'(t)=0 (11)
and for a phase detection error .phi. EQU I'[t]=Cos [.phi.]Is'[t] (12) EQU Q'[t]=-Sin [.phi.]Is'[t] (13)
This gives: EQU Is'[t]=Sqrt[I'[t] 2+Q'[t] 2] (14)
so that: EQU Cos [.phi.]=I'[t]/Is'[t]=I'[t]/Sqrt[I'[t] 2+Q'[t] 2] (15) EQU Sin [.phi.]=-Q'[t]/Is'[t]+-Q'[t]/Sqrt[I'[t] 2+Q'[t] 2] (16)
Is[t] and Qs[t] are found in terms of I[t], Q[t], Cos[(.phi.] and Sin[.phi.] to get: EQU Is[t]=Cos [.phi.]I[t]-Sin [.phi.]Q[t] (17) EQU Qs[t]=Sin [.phi.]I[t]+Cos [.phi.]Q[t] (18)
which may be rewritten as: EQU Is[t]=(I'[t]I[t]+Q'[t])/Is'[t] (19) EQU Qs[t]=(-Q![t]I[t]+I'[t]Q[t])/Is'(t) (20)
providing Is[t] and Qs[t], the synchronous in-phase and quadrature outputs as required.
The problem with this technique is that the Nyquist slope of the incoming signal, as shown in FIG. 5, is characterized by: ##EQU1## where .omega..sub.t is the deviation frequency (the angular frequency .omega. is related to frequency f in sampled systems by .omega.=2 .pi.f/f.sub.s where f.sub.s is the sample frequency). For NTSC and PAL countries f is 0.75 and 1.0 Mhz respectively. Therefore a deviation from the center frequency of only a few percent makes the response of this system asymmetrical enough to distort the response on the assumption of a double-sideband signal.
Essentially the system becomes the sum of a double sideband system (the smallest sideband of frequency .omega..sub.m -.omega..sub.l and w.sub.m +.omega..sub.l) with a single sideband system (the difference between the largest and smallest sidebands). If .omega..sub.t is positive, then EQU A[.omega..sub.m .omega..sub.l ]=1-.omega..sub.l /.omega..sub.t(22) EQU A[.omega..sub.m .omega..sub.l ]=1+.omega..sub.l /.omega..sub.t(23)
so the double sideband component is EQU Double[.omega..sub.l ]=1-.omega..sub.l /.omega..sub.t (24)
and the single sideband component is EQU Single[.omega..sub.l ]=2.omega..sub.l /.omega..sub.t. (25)
Ideally a filter should maximize .omega..sub.t (thereby Double[.omega..sub.l ]=1 and Single[.omega..sub.l ]=0) by flattening the overall frequency response around .omega..sub.m. The compensating filter is defined as the reciprocal of the signal response above (the region defining the response around (.omega.=.omega..sub.m) EQU A[.omega.]=2/(1+(.omega.-.omega..sub.m)/.omega..sub.t). (26)
However this filter has an infinite amplitude response around .omega.=(.omega..sub.m -.omega..sub.t), which is not possible with a finite impulse response filter (which gives the best performance in terms of the signal delay as a function of .omega.). Therefore the active range of .omega..sub.l is limited to .omega..sub.m .+-..omega..sub.d, where .omega..sup.d &lt;.omega..sub.t.
Thus the input to the compensation filter is limited so that the response outside of .omega..sub.m .+-..omega..sub.d is rendered insignificant, and the filter, as shown in FIG. 6, is designed to give the desired response over the range .omega..sub.m .+-.-.omega..sub.d. The pre-filter that does the limiting is designed with well-known FIR filter design techniques. However traditional designs for the compensation filter often result in very ill-conditioned design equations that usually yield filters with large coefficients.
What is desired is a technique for designing digital FIR filters for compensation of Nyquist slope in pseudo-synchronous demodulators to obtain an approximately flat overall response over a narrow frequency range around the 1/2-amplitude response point of a Nyquist filter which reduces the ill-conditioned design problems, resulting in considerably less dynamic range requirement for the filter arithmetic.